1 INTRODUCTION

1.1 Motivation In case of a failure of flood protection measures (e.g. dikes, mobile walls) a wave is initiated into the hinterland and may cause extensive damage in densely populated floodplains. Recurring dike breaks at massive flood events indicate that inland flood protection is vulnerable and the resulting risk has to be assessed. A Risk Assessment proce-dure has been developed at the Institute of Hy-draulic Engineering and Water Resources Man-agement (IWW), RWTH Aachen University. The basic principle of this approach, which was origi-nally developed for German dams by Rettemeier et al. (2000), can be transferred to rivers and dikes (Kutschera et al., 2008). Mathematically, the risk is obtained by multiplying the probability of a dike failure by the extent of the damage caused in the event of a collapse. To assess the risk pro-voked by floods and corresponding protective

structures, the potential damage on people and property has to be determined, which involves the identification of inundated areas as well as flow depths and velocities of the initiated wave.

The static impact basically depends on the total water volume entering the hinterland over a long period. This state has to be considered when fo-cussing on the long-term and large-scale inunda-tion in the entire floodplain. The steady-state breach discharge through a collapsed dike section significantly affects final water levels and the ex-tent of inundated areas, while during the first tran-sient phase the combination of flow velocities and water depths within the dike-break wave induces dynamic damages nearby the breach location. Compared to the duration of the whole flood event this period is often short but equally dangerous for people and property in the affected region.

The results of dike-break induced flow compu-tations are used to identify and illustrate areas of significant flood risk (flood mapping). Simula-tions of scenarios may also provide authorities

Hybrid modelling of dike-break induced flows

S. Roger, J. Köngeter & H. Schüttrumpf Institute of Hydraulic Engineering and Water Resources Management (IWW), RWTH Aachen University, Hydraulic Laboratory Kreuzherrenstraße 7, 52056 Aachen, Germany

S. Erpicum, P. Archambeau & M. Pirotton

Department ArGEnCo, Research unit of Hydrology, Applied Hydrodynamics and Hydraulic Constructions (HACH), University of Liege, Chemin des Chevreuils 1 Bât. B52/3+1, 4000 Liege, Belgium

D. Schwanenberg Deltares, Operational Water Management, Rotterdamseweg 185, 2629 HD Delft, NL

B.J. Dewals Belgian Fund for Scientific Research F.R.S.-FNRS

ABSTRACT: In a hybrid approach experimental model data are combined with results from 3D and 2D numerical modelling. The latter was conducted by two different models solving the depth-averaged shal-low water equations. 3D computations are based on the REYNOLDS-averaged NAVIER-STOKES equations (RANS) using a volume of fluid approach to capture the free water surface. Measurements were per-formed on a scale model which was especially designed to reproduce the specific conditions of dike breaks. In various simulations it turned out that 2D shallow water models are able to reproduce steady-state flow patterns of dike-break induced flows and that there is a low sensitivity of the solution concern-ing turbulence modelling, bed and wall roughness. Nevertheless, final flow splits and breach discharges are systematically underestimated. This discrepancy seems to result from inherent modelling assumptions such as zero-vertical velocity and hydrostatic pressure distribution. Therefore, the complementary use of 3D RANS and 2D depth-averaged modelling frameworks for detailed predictions of dike-break induced flows is discussed in the present paper, based on BOUSSINESQ and pressure coefficients, which represent effects of non-uniform velocity profiles and non-hydrostatic pressure distribution over water depth, re-spectively. Values of these coefficients are inferred from 3D numerical results for the final steady state.

Keywords: BOUSSINESQ coefficients, Discontinuous Galerkin, Finite volume, Dike break, Hybrid model-ling

River Flow 2010 - Dittrich, Koll, Aberle & Geisenhainer (eds) - © 2010 Bundesanstalt für Wasserbau ISBN 978-3-939230-00-7

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with valuable information like flood arrival time or main flow directions to manage the residual risk: definition of evacuation zones, coordination of civil protection, and land use planning.

1.2 Phenomenon In contrast to the wide knowledge of flood wave propagation initiated by dam breaks (Hervouet & Petitjean 1999, Brufau & Garcia-Navarro 2000, CADAM 2000, Toro 2001, RESCDAM 2001, IMPACT 2005) the expertise of dike-break in-duced flows is not satisfactory. The latter are in-fluenced by the main flow direction of the river parallel to the flood protection structure (asym-metric flood wave propagation). Moreover, unlike reservoirs at rest, a river bed will not be empty: the persisting flood discharge of the river leads to a fixed water level in the breach after a certain time (final steady state) resulting in a partition of the inflow into the downstream and the breach discharges, which partly corresponds to a with-drawal of water. A dike break may thus be classi-fied between a dam break and a channel junction as illustrated in Figure 1.

The related flows can be characterized as large-scale, long-term but at the same time transient, 3D, turbulent and multiphase (air entrainment, se-diment transport) flows with complex free-surface behaviour upon a heterogeneous topography (in-itially dry, geometrically complicated, covered by buildings and plants).

There is a lack of knowledge as regards these types of flood waves. The existing measured data are not sufficient due to the unpredictability, rare-ness and the danger of such events. The safety of threatened people and property as well as the se-curity of the gauging staff during the event has priority, which complicates complex measuring campaigns close to a breach.

There exist only few investigations considering the propagation of a wave into an area, such as Fraccarollo & Toro (1995), Stelling & Duinmeijer (2003), Kulisch (2002), Shige-eda & Akiyama (2003), Aureli et al. (2004), and Eaket et al. (2005) relating to dam-break flows, however. Au-reli & Mignosa (2002, 2004) only considered the presence of a permanent river discharge. That is why numerical models can not be properly vali-dated and therefore may not calculate reliable simulation results for complex scenarios.

1.3 Hybrid procedure Circumventing the expense of a full-sized proto-type, a bench-scale model was used to provide experimental data, which were recorded with so-phisticated measurement techniques to explore

flow effects and to validate numerical models. In an experimental model natural phenomena are scaled down and idealised resulting in differences between modelled and real procedures. A numeri-cal model contains simplifications in the mathe-matical description and numerical treatment of flow phenomena. To solve this problem, experi-mental and numerical models are combined in a hybrid approach to benefit from each other by ex-ploiting their respective advantages and compen-sating their weak-points at the same time.

Figure 1. Classification of dike-break induced flows.

On the one hand, the accuracy of numerical forecasts can directly be quantified by measure-ments. On the other hand, numerical simulations complement the model tests by calculating scena-rios of different configurations, geometries and boundary conditions. The combination enables se-lective improvements in numerical methods and more reliable forecasts for long-term and large-scale applications. The forecast quality is directly quantifiable by comparisons between simulation results (coming from different numerical schemes and mathematical approaches) and measured data in terms of flow discharges, water depths, flow velocities, and arrival times (depending on the fact whether the first dynamic phase or the final steady state is evaluated).

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Figure 2. Scale model set-up.

2 EXPERIMENTAL MODEL

2.1 Apparatus The idealised experimental set-up in Figure 2 takes into account specific boundary conditions of dike-break induced flows close to the breach sec-tion (Briechle & Köngeter, 2002, Briechle et al., 2004). It consists of a horizontal rectangular channel (width 1 m) with a pneumatically driven gate at one bank and an even adjacent flat propa-gation area (3.5 times 4.0 m²) made of glass. The entire opening of the flap gate takes less than 0.3 s, which represents the worst case scenario of a sudden and complete dike failure. Moreover, the opening mechanism is a combination of pull and rotation to minimize influence on the free water column when the wave is initiated. As opposed to straight flumes, the water propagates radial and falls off the glass plate freely at three edges. The bottom of the propagation area is made of glass to

minimize roughness effects and to enable laser measurements from below the plate. Initial water levels in the channel (0.3 – 0.5 m), channel dis-charges (0.1 – 0.3 m³/s), and breach widths (0.3 – 0.7 m) could be varied.

2.2 Measuring techniques Due to strong temporal and spatial variations of the initiated wave as well as air entrainment, an advanced non-intrusive measuring technique was necessary providing high frequencies and stability towards dynamically changing water levels. Thus, water depths were recorded by ultrasonic sensors with 25 Hz frequency all over the propagation area with grid lengths of Δx = Δy = 0.2 m, and a refined grid of 0.1 m close to the breach zone. Within the channel detection was performed at different cross sections. Up to eight sensors were mounted on movable cross-beams enabling meas-urements at various locations.

Vertical velocity profiles u(z), v(z) were sam-pled using a conventional 1D Laser-Doppler

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Anemometer (LDA), mounted on an automatic traversing unit beneath the glass plate. Horizontal velocity components over depth are sampled at three cross-sections y = 0.25, 0.30, 0.35 m near the breach in a dense grid of Δx = 0.05 m, Δz = 0.01 m within the wave. These profiles pro-vide a distribution of momentum correction coef-ficients (BOUSSINESQ coefficients), which may be exploited for 2D numerical modelling.

2.3 Test configurations Four initial hydraulic configurations (Table 1) were considered differing in the discharge at the channel inlet and in the initial channel flow depth. A steady flow was established in the channel, be-fore the gate was opened in each model run. Then, the inflow splits into the breach discharge and the weir overflow from the downstream channel.

Table 1. Test configurations for breach width b = 0.70 m. ______________________________________________ Test ID Inflow Initial depth Crest height [l/s] [cm] [cm] ______________________________________________ Q300-h50 300 50 24.1 Q300-h40 300 40 15.2 Q200-h50 200 50 29.7 Q200-h50 200 40 20.2 _____________________________________________ The channel inflow was controlled via an ultra-sonic flow-measuring device. A weir at the chan-nel end was calibrated for different crest heights to control the initial water depth. The rating curve for each of those weir positions and the corre-sponding discharge coefficients were determined experimentally running various known discharges from the inlet while keeping the gate closed. Measuring the water level at the channel end yields the discharge over the weir. Consequently, the steady-state breach discharge QB was indi-rectly calculated as the difference between model inflow and weir overflow Qw. Figure 2 also shows the different boundary conditions.

3 MATHEMATICAL AND NUMERICAL MODELS

3.1 Shallow water models Firstly, numerical simulations of the experimental flow configurations have been performed with two different 2D models solving the depth-averaged Shallow-Water Equations (SWE). The basic assumption states that velocities normal to the main flow directions remain small. As a con-sequence the pressure field is hydrostatic, which limits the applicability of the SWE.

On the one hand DGFLOW is applied, a total variation diminishing (TVD) RUNGE KUTTA dis-continuous GALERKIN (RKDG) finite element (FE) method on unstructured triangular meshes, developed at IWW, RWTH Aachen University (Schwanenberg & Harms, 2002, 2004). On the other hand WOLF 2D is used, a finite volume (FV) scheme involving a flux vector splitting (FVS) method on a multiblock structured grid, developed at the University of Liege (Erpicum et al., 2010a).

Both schemes are well suited to handle tran-sient, rapidly varying dam-break induced flows with steep gradients over a dry bed. Now they are applied to the dike-break induced flow conditions in the experimental test arrangement. As Roger et al. (2009) describe the computational methods in detail the two following paragraphs and Table 2 only delivers a brief insight.

Table 2. Comparison of the two applied SWE models DGFlow and WOLF 2D. ______________________________________________ Characteristics DGFlow WOLF 2D ____________ ___________________________________________________________ Model type Finite Element Finite Volume Space Discontious Flux Vector Discretisation GALERKIN (DG) Splitting (FVS) ______________________________________________ Time integration TVD RUNGE-KUTTA RUNGE-KUTTA Numerical grid Triangular Cartesian Number of elements 23,000 60,000 Edge lengths [m] 0.03 – 0.05 0.02 Bottom friction MANNING MANNING Turbulence Closure None Algebraic/k-ε Wall roughness None Accounted for BOUSSINESQ coeff. Set to unity Distributed values _____________________________________________

3.2 DGFlow DGFlow is based on the RKDG method for hy-perbolic equation systems and the local discon-tinuous Galerkin method for advection-dominated flows (Cockburn 1999, Cockburn et al. 2000). The leadoff implementation of the RKDG method to the SWE was presented by Schwanenberg and Köngeter (2000). The scheme can be divided into three main steps: − Decoupling the partial differential equations by

a DG space discretisation (polynomial degree k) into a set of ordinary differential equations

− Integrating the ordinary differential equations in time by a (k+1)-order TVD RK method

− Applying a slope limiter on every intermediate time step introducing a selective amount of dis-sipation to obtain stability at shocks

3.3 WOLF 2D WOLF 2D solves the divergence form of the shal-low-water equations by means of a finite volume

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scheme. Variable reconstruction at cells interfaces is either constant or linear, combined with a slope limiter, leading in the latter case to second-order space accuracy. The fluxes are computed by a self-developed Flux Vector Splitting (FVS) method, in which the upwinding direction of each flux is simply dictated by the sign of the flow ve-locity reconstructed at the cell interfaces.

A “VON NEUMANN” stability analysis has dem-onstrated that this FVS leads to a stable spatial discretization, requiring low computational cost. This FVS offers the advantages of being com-pletely Froude-independent and of facilitating a satisfactory adequacy with the discretization of the bottom slope term. The turbulent fluxes are simply evaluated by means of a centred scheme.

The time integration was performed here by means of a 3-step first-order accurate RK algo-rithm. A semi-implicit treatment of the bottom friction term in was used, without requiring addi-tional computational cost.

Besides, wetting and drying of cells is handled free of mass conservation error by means of an it-erative resolution of the continuity equation at each time step. A grid adaptation technique re-stricts the simulation domain to the wet cells.

This finite volume scheme has already proven its validity and efficiency for numerous applica-tions (Dewals et al., 2008, Erpicum et al., 2009, Erpicum et al., 2010b).

3.4 Star-CD Subsequently, a fully 3D model has been set up: the commercial code Star-CD (CD-adapco, 2005) is based on the REYNOLDS-averaged NAVIER STOKES equations (RANS) and accounts for the free surface location via a volume of fluid method (Table 3).

Table 3. Main characteristics of RANS model Star-CD. ______________________________________________ Model type Finite Volume Space Discretisation Upwind Differencing Scheme ______________________________________________ Time integration Implcit scheme Numerical grid Triangular prisms Number of Cells 250,000 – 300,000 Cell dimensions [m] 0.02 – 0.05 m Bottom friction wall function Turbulence Closure k-ε-model Wall roughness wall function _____________________________________________

Roughness and turbulence effects are consid-

ered via a wall-functional approach and a k-ε-turbulence model, respectively. Once again the different scenarios of the scale model tests are computed in 3D. All results are compared with the measurements and exploited in detail in terms of flow depths, flow velocities, and calculation of the

breach discharge. In some cases (dispersion, dis-sipation), the 3D information is used to estimate appropriate settings for the 2D approaches.

4 NUMERICAL MODELLING

4.1 Computational procedure and former results All numerical simulations of the different experi-mental configurations reproduce the basic flow pattern satisfactorily (Figure 3), which confirms the convergence and general applicability of the methods on dike-break type of problems. At first the initial state was computed for a closed flap gate which corresponds to a simple steady channel flow with a sharp crested weir at the outflow sec-tion. According to the opening mechanism in the scale model tests, an immediate and complete failure of flood protection measures with a rectan-gular breach was realised in the numerical compu-tations by changing the respective boundary con-dition at the breach location. Simulations contained the transient propagation and develop-ment of the dike break wave in the floodplain as well as the open channel flow. The whole flooding event was modelled, starting from an initial condi-tion until the final steady solution in a single sta-ble run without oscillations.

The resulting deviations from the performed measurements were analysed focussing on the simplifications of the mathematical model or the numerical approach (Roger et al., 2009). The lat-ter has to deal with complex free surface struc-tures or steep gradients. As regards the mathe-matical models the different simulations enable direct comparisons between the assumptions of the SWE and the less simplified RANS in relation to the collected experimental data.

In addition to the hydraulic (discharge, initial water level) and geometric (breach width) bound-ary and initial conditions, several numerical pa-rameters were examined. A preliminary study with DGFlow has yielded the required fineness of the numerical grid in order to obtain an almost mesh independent solution and to minimize the discretization error (Roger et al., 2009). A sensi-tivity analysis was performed as regards bed roughness. Additional simulations were performed with WOLD 2D introducing non-uniform BOUS-SINESQ-coefficients (for uneven horizontal veloc-ity distribution over depth) as well as different turbulence closure schemes and wall roughness coefficients to analyze whether these approaches affect the computed 2D solutions compared to the measurements.

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a) Experimental dike break wave.

b) Numerical dike break wave (Star-CD).

Figure 3. Experimental (a) and numerical (b) modelling of dike-break induced flow.

4.2 Boundary conditions At the channel inlet (Figure 2), the unit discharge was prescribed as an upstream inflow boundary condition. At the edges of the propagation area (glass plate) no boundary condition was needed for the obtained supercritical outflow.

Specific difficulties occur with respect to the modelling of the flow split into breach discharge and the discharge in the downstream channel. Par-ticularly, in combination with the downstream boundary condition, this additional degree of freedom may lead to numerical problems (Roger et al., 2006). In both 2D codes the calibrated weir formulas for different crest heights (Table 1) were implemented as possible downstream boundary conditions (Roger et al., 2009) while the complete weir has been geometrically discretized for the 3D computations. Thus, the dynamics of the weir dis-charge depending on the actual channel water level could be modelled properly. The resulting backwater effects interact with the flow resistance of the downstream channel reach compared to the breach section as well as with the flow split into breach discharge and downstream channel dis-charge. The 90°-turn of momentum from the main channel flow to the breach main axis is closely re-

lated to the interplay of flow partition and down-stream boundary.

4.3 Breach discharge The large-scale and long-term inundation of a floodplain in practical applications is strongly af-fected by the steady-state discharge through the breach. Thus, its correct calculation is the decisive parameter to evaluate model modifications. Simu-lated and measured discharges across the breach are compared in Table 4 and Fig. 4. Though the qualitative flow split for all four configurations is modelled correctly, the 2D-numerical predictions underestimate the discharge released into the floodplain by 6 to 11% of the test breach dis-charge. Consistently with theoretical hypothesis, the 3D results fit better the scale model data than the shallow water approaches, without succeeding in exactly calculating the breach discharge. The next paragraphs aim at investigating whether this issue can be enhanced in the framework of depth-averaged flow modelling.

Table 4. Measured and computed breach discharges. ______________________________________________ Test ID Measured DGFlow WOLF 2D Star-CD [l/s] [l/s] [l/s] [l/s] ______________________________________________ Q300-h50 218 200 198 211 Q300-h40 159 141 141 153 Q200-h50 194 183 182 190 Q200-h50 154 143 141 148 _____________________________________________

0.0

50.0

100.0

150.0

200.0

250.0

Q300-h50 Q300-h40 Q200-h50 Q200-h40

Bre

ach

disc

harg

e (l/

s)

Measured DGFlowWOLF 2D Star-CD

Figure 4. Measured and computed breach discharges.

For test configuration Q300-h50, the simulations were repeated with two types of turbulence clo-sures implemented in WOLF 2D, that is a purely algebraic expression for eddy viscosity and a two-length-scale depth-averaged k-ε model. Moreover, the influence of bed and wall roughness was in-vestigated. As described in Roger et al. (2009) and summarized in Table 5, the turbulence closure as well as bed and wall roughness have basically a minor effect on the computational results.

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Table 5. Influence of turbulence and wall-roughness on breach discharges predicted by WOLF 2D. ______________________________________________ Test ID Measured WOLF 2D WOLF 2D WOLF 2D [l/s] [l/s] Turb[l/s] Wall[l/s] ______________________________________________ Q300-h50 218 198 198 202-204

4.4 Velocity profile and pressure distribution The 6 to 11% discrepancy between measured val-ues and the breach discharges simulated based on the 2D depth-averaged model is found to remain essentially insensitive to the turbulence closure and the roughness coefficients. This suggests that the difference may result from the basic assump-tions inherent to depth-average modelling, such as simplified velocity profile and hydrostatic pres-sure distribution along the water depth. This was investigated based on the evaluation of BOUSSI-NESQ coefficients ρxx, ρyy, ρxy and pressure coeffi-cient Λ, defined as follows:

( ) ( )

2 22 2; ;

; 0.5xx yy

xy

u u v v

uv u v p gh

ρ ρ

ρ ρ

= =

= Λ = + (1)

where h = flow depth, u, v = local velocity com-ponents in x- and y-directions, p = water density, g = gravitational acceleration, ρ = water density and angle brackets representing depth-averaging.

The BOUSSINESQ coefficients are all equal to unity in case of a uniform velocity profile along the water depth, whereas the greater the BOUSSI-NESQ coefficients the more irregular the velocity profile. In standard shallow-water models, BOUS-SINESQ coefficients are all implicitly set to unity. It may be mathematically demonstrated that the value of ρxx and ρyy, is never below unity, while ρxy verifies: ρxy ρxy < ρxx ρyy. In river flows, BOUSSINESQ coefficients remain usually very close to unity, except if significant secondary flows occur (e.g. in bends). Similarly, the pressure coefficient equals unity in case of hydrostatic dis-tribution and deviates from unity in all other cases.

Here, the velocity and pressure profiles along the water depth computed by Star-CD were used to evaluate the BOUSSINESQ and pressure coeffi-cients by integrating manually the horizontal ve-locities and the pressure values over water depth. As detailed in Fig. 5, the pattern of BOUSSINESQ coefficients ρxx reveals mainly two regions of sig-nificantly non-uniform velocity profiles, namely in the dike-break wave and at the beginning of the downstream channel reach. The former probably has no influence on flow split because of the su-percritical flow in advance. The latter, however, may strongly affect the discharge partition. Con-sidering the 2D-results, the flow resistance of the

downstream channel reach tends to be too low compared to the breach section. That might be the reason why too much water goes over the weir in-stead of entering the floodplain.

Figure 5. BOUSSINESQ coefficient ρxx.

Figure 6. Pressure coefficient Λ.

The pattern of BOUSSINESQ coefficient ρxy in the downstream channel is found similar to the pattern of ρxx represented in Fig. 5. Hence, ρxy re-flects a non-uniform profile of flow velocity which also contributes an increase to the effective flow resistance in the downstream channel. This effect is disregarded in the 2D depth-averaged predictions of breach discharge. Although the third BOUSSINESQ coefficient ρyy also takes values significantly deviating from unity in the channel, it may not be invoked as an explanation for the underestimation of breach discharge. These high values of ρyy simply result from zero-crossings of the very low depth-averaged transverse velocity components throughout the whole channel.

Finally, the pressure coefficient Λ is represented in Fig. 6. The pressure is found to show an interesting pattern of secondary waves superimposed on the main dike-break wave. Again, this effect does not play a part in the flow split since it is located downstream of the transi-tion from subcritical to supercritical flow. But in a confined region of the channel there are devia-tions from hydrostatic distribution which also may enlarge the flow resistance for the main channel flow.

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5 CONCLUSIONS

Dike-break induced flow has been investigated based on hybrid modeling, involving physical modeling as well as 2D depth-averaged and com-plete 3D simulations. While the 2D simulations were shown to underestimate the breach dis-charge, post-processing of 3D results has been successfully used to demonstrate that non-uniform velocity profiles as well as non-hydrostatic condi-tions near the breach in the channel are most probably the main reasons for this underestima-tion. Additional simulations introducing a reliable spatial distribution of BOUSSINESQ and pressure coefficients may clarify their respective impact.

ACKNOWLEDGEMENTS

The first author thanks the German Research Foundation (DFG) for funding his research project under reference number KO 1573/15-2.

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